Cell phones work by sending and receiving radio waves. Mirrors as we know them are materials that reflect EMR in the visible region, not radio waves. Radio waves can go straight through mirrors, so you could make a phone call.

Let's consider very reflective materials, like silver or aluminium. When they are used for mirrors (i.e. the really shiny silver/aluminium) they are in essence very large crystals in the solid state (their atoms are arranged in large, structured molecules). What you end up with is a material that has a so-called "sea of electrons" on its surface such that the electrons are very mobile. This also means all the different states of the electrons available are extremely diverse and can interact with many types of light. In the case of aluminium and silver this is visible light especially. When the light interacts with the surface it tends to be reflected rather than absorbed. There are many quantum effects that cause mirrors to make it only reflect EMR in the visible region (having to do with valence configuration of the metal, size of the atoms etc.), so I'll leave that to someone else explain that better than I.

Reflective substances in general have this property of being able to interact with many forms of light, and more importantly are structured in an orderly fashion to reflect the light with high fidelity, and not just scatter it. A very flat, even surface allows this. That's why mirrors have a very thin layer of extremely uniform, flat silver or aluminium backing, the rest is just glass to protect it from oxidation.

The heat of sunlight comes from the infrared region, which is just adjacent to the visible region, so I wouldn't be surprised if it does reflect some of this heat.

Different materials do different things, so it gets to be educated trial and error thing for macroscopic systems (like making mirrors) but can get more controlled for smaller systems (e.g. quantum dots, which are nanoparticulate semiconductors, easier from a theoretical perspective to manipulate how they interact with light). As an example of how different materials do different things, consider the "windows" on an X-ray tube found in your friendly neighbourhood X-ray machine. While the ports appear opaque, they are in fact transparent to X-rays, allowing fantastic vacuum containment while still allowing them to transmit X-rays which tend to interact with anything larger than a few atoms big. Why is this? Because the window is covered with a thin foil of beryllium, whose atoms are very small and don't interact much with X-rays.